Resummation of Large Logs in DIS at x->1 Xiangdong Ji University of Maryland SCET workshop, University of Arizona, March 2-4, 2006

Outline Introduction to DIS at large x and resummation of large logarithms Resummation to N 3 LL in the standard and EFT approaches Puzzles in SCET factorization Cancellation of the spurious scale Summary

Inclusive DIS Consider the text- book example of inclusive DIS on a proton target As Q->∞, at a fixed Bjorken x, the process can be factorized, as shown by the reduced diagram on the right.

QCD factorization Standard QCD factorization for DIS f is the parton distribution function, nonperturbative C is the coefficient function, a power series in coupling α s As Bjorken x->1, the pQCD series converges slowly A resummation is needed to get reliable predictions

Physical origin As x-> 1, the hadron final state has an invariant mass Q 2 (1-x), which becomes an independent scale. Thus, the hadron final state is restricted to a hadronic jet plus arbitrary number of soft gluons radiations. Soft gluons contribution is usually large due to infrared enhancement near the edge of phase-space. One must sum these soft gluons, just like in the case of QED where one must sum over soft photon contributions when the detector resolution is high (large logarithms).

In moment space In moment space, the factorization becomes The expansion parameter is α s ln 2 N!

Exponentiation The large logarithms exponentiate! A property obvious easily seen in QED. In QCD, it requires some additional study of color factors, The expansion parameter is now α s lnN!

Resummation Consider α s lnN is of order 1, sum over all terms of same order in α s such α s lnN, ( α s lnN )2, ( α s lnN) 3, etc where =  0 α s lnN. The expansion is now in α s We need to find what g n ( ) are g 1 ( ): Leading Logarithms (LL) g 2 ( ): Next-to-Leading Logarithm (NLL) g 3 ( ): Next-to-Next-to-Leading Logarithm (N 2 LL)

Sterman’s approach Re-factorization of the DIS structure function at new scale Q 2 (1-x). Introducing new ingredients such as jet functions, soft factor, and real hard contribution Write done (complicated) differential equations for jets and soft factor at large x, which when solved yield exponentiated x-dependence.

Result A is the anomalous dimension of a Wilson-line cusp A=  α s n A n B is a perturbation series B=  α s n B n which can be extracted from fixed order calculation LL: A 1 NLL: A 1,A 2,B 1 N 2 LL: A 1 -A 3,B 1,B 2 N 3 LL: A 1 -A 4,B 1 -B 3

Resummed functions Up to N 3 LL, all are known except A 4

An EFT Approach A. Manohar, Phys. Rev. 68, (2003) Based on SCET, conceptually simple and readily generalizable to other processes. Result obtained to NLL, agrees with old approach Improvements and to N 3 LL (Idilbi, Ji, Ma and Yuan, hep-ph/ ) Take Q->  first and (1-x) is small but not correlated with Q. An actual formulation of effective field theory, such as SCET is entirely unnecessary. Result agrees with the old one to all orders in principle, and to N 3 LL explicitly.

EFT Approach in a nutshell Main idea: integrating out physics at different scales stepwise and connecting different scales using renormalization group running. Main steps: Integrating out physics at scale Q 2 by matching to effective current Taking care of physics between Q 2 and Q 2 (1-x) by RG running of the effective current Integrating out physics at scale Q 2 by matching to parton distribution function RG running of PDF through DGLAP

Matching at Q 2 At scale Q, one can integrate out perturbative physics from virtual gluons in the vertex type of diagrams,

Running from Q 2 to Q 2 (1-x) The physics between scale Q 2 to Q 2 (1-x) can be taken care of by solve the renormalization group equation for the scale evolution of the effective current Where B is the related to the coefficient of the delta function in the anomalous dimension

Matching at Q 2 (1-x) At this scale, one must consider soft gluon radiations. Integrating out these radiations matches the theory to parton distributions. The calculation is exactly the same as in the full QCD, therefore, one can take the full QCD result in the soft-collinear limit, where the logarithms of type lnQ/N  has been set to zero

Final Result in EFT Put all factors together some additional manipulation shows the full equivalence with the traditional approach. Comments No actual EFT is needed! Only new scale Q 2 (1-x) appears, which is assumed to be perturbative. Power counting in 1-x. Resummation is entirely accomplished. Conceptually much simpler than original approach.

How does one connect the EFT approach to Sterman’s approach?

Need an actual formulation of EFT Is it SCET? Maybe: Expansion parameter (1-x) is can be identified as SCET expansion parameter 2 = (1-x) «1 Maybe Not: In the usual resummation, (1-x) α Q » Λ QCD, for any α >0. In SCET, Q is usually ~ Λ QCD. Thus SCET is defined in a very small kinematic region, whereas the usual resummation works in a much wider region. In this limit kinematic region, SCET may or may not generate the correct resummation, because the scale Q is generally non-perturbative.

Questions over SCET Factorization B. D. Pecjak, JHEP10 (2005) 040. Non-factorizable contribution to DIS at large x In principle, this is not a problem because there is no proof that the DIS in this region is factorizable. J. Chay & C. Kim, hep-ph/ There is a non-perturbative soft contribution in additional to the usual parton distribution. Soft contribution is at scale Q(1-x) and is non-perturbative. A different factorization and hence the resumed perturbative part is different from the usual coefficient function.

SCET factorization In the second stage matching, one can obtain a SCET factorization by matching the DIS process in SCET I on to a product of jet function, soft factor and parton distribution Chay & Kim

Puzzles Jet functions reproduces entirely the matching at Q 2 (1- x) There is no room for the soft contribution

Role of soft function? Explicit calculation shows that the soft factor has no infrared divergence and lives in the scale Q(1-x) which is on the order of Λ QCD Only in that sense the soft factor is non-perturbative! New factorization beyond the usual pQCD factorization?

Scale cancellation? Thus, SCET factorization is in principle outside of the usual pQCD factorization range. Since the coefficient function is at the scale Q 2 (1-x), thus the physics in the soft factor must be cancelled by that in the jet function and parton distributions. Therefore the non-pert. scale Q(1-x) in SCET is spurious: although it is non-perturbative, but its dependence cancels. Similar scale cancellation may happen for the calculation of Pecjak, in a way more subtle than that suggested by A. Manohar.

Summary Using EFT concepts, resummation of large logs in DIS at large x can be done very simply using the renormalization group approach. (Now to N 3 LL) SCET factorization of DIS at large x introduces a new small scale Q(1-x). However, this scale cancels in the product. Thus, the DIS resummation works even when Q(1-x) is on the order of Λ QCD SCET factorization is not the most efficient way to characterize the important regions of momentum flow.

Generating large-x partons Large x-partons are generated through soft-gluon radiation One can write done a differential equation for large-x parton distribution Knowing the kernal, the solution can be written formally as

Large-x jet function In the large-x region, the jet function satisfy the following equation

Solution of the equation

Explicit form of factorization

In moment space Large double logs